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Discussion questions on systems of linear equations

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Please provide an answer to the following questions below that contains 250 to 300 words each.

1.By looking at two linear equations, how can you tell that the corresponding lines are parallel, the same graph, or intersecting lines? How many solutions does each possibility have and why is that? Show examples for each possible situation.

2.Is there a difference between solving a system of equations by the algebraic method and the graphical method? Why? What are the advantages and disadvantages of each?

3.Write 250-300 words comparing and contrasting all methods of solving systems of linear equations with two variables. Explain which method you prefer and why. Support your answer by appropriate examples.

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Solution Preview

Hi,

Please find the solutions/explanations attached herewith.

Please provide an answer to the following questions below that contains 250 to 300 words each.

1. By looking at two linear equations, how can you tell that the corresponding lines are parallel, the same graph, or intersecting lines? How many solutions does each possibility have and why is that? Show examples for each possible situation.

Solution:

PARALLEL LINES:

If two equations have same slopes then the lines are parallel.

Let us take example

y = 3x + 7

and y = 3x - 5

We can see that the slope of both the lines is 3. Therefore, lines are parallel.

These types of system of equations have NO Solution as they don't have any common point which satisfies both the equations.

SAME GRAPH:

The equations have same graph when both the equations are same.

Let us take one example to illustrate it.

x + y = 2
2x + 2y = 4

We can see that we can get second equation by multiplying first equation by 2.

Or just divide second equation by 2, we will first equations, therefore they have same graph. These types of system of equations have infinity many solutions as if we give any different values to one variable we will get corresponding for other variable.

INTERSECTING LINES:

If the system of linear equations are not of the above two types then the lines are intersecting lines. That is if neither the equations are of parallel lines nor have same graph then the lines are intersecting lines.

Let us take the example
x + y = 1
x - y = 3

The slope of first line is -1 and slope of second line is 1. ...

Solution Summary

Detailed explantions to all the quesions is provided.

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See Also This Related BrainMass Solution

Graphing and linear equations

Please see attached for the complete details of the given problems:

WEEK HOMEWORK
Please add explanation and colored fonts on your answers

Page 483, #2

Solve by graphing. Indicate whether each system is independent, inconsistent, or dependent.

Page 483, #6

Solve by graphing. Indicate whether each system is independent, inconsistent, or dependent.

Page 483, #8

Solve each system by the substitution method. Indicate whether each system is independent, inconsistent, or dependent.

Page 483, #10

Solve each system by the substitution method. Indicate whether each system is independent, inconsistent, or dependent.

Page 483, #14

Solve each system by the substitution method. Indicate whether each system is independent, inconsistent, or dependent.

Page 484, #18

Solve each system by the addition method. Indicate whether each system is independent, inconsistent, or dependent.

Page 484, #20

Solve each system by the addition method. Indicate whether each system is independent, inconsistent, or dependent.

Page 484, #24

Solve each system by the addition method. Indicate whether each system is independent, inconsistent, or dependent.

Page 485, #47

Perimeter of a rectangle. The length of a rectangular swimming pool is 15 feet longer than the width. If the perimeter is 82 feet, then what are the length and width?

Page 485, #48

Household income. Alkena and Hsu together earn $84, 326 per year. If Alkena earns $12,468 more per year than Hsu, then how much does each of them earn per year.

Individual Textbook Problems

Page 459, #20

Solve each system by graphing.

and

Page 461, #70

Solve each system by the substitution method.

and

Page 461, #92

Investing her bonus. Donna invested her $33,000 bonus and received a total of $970 in interest after one year. If part of the money returned 4% and the remainder 2.25%, then how much did she invest at each rate?

Page 471, #52

Solve each system by substitution or addition.

2y - x = 3
x = 3y - 5

Page 472, #72

Books and magazines. At Gwen's garage sale, all books were one price, and all magazines were another price. Harriet bought four books and three magazines for $1.45, and June bought two books and five magazines for $1.25. What was the price of a book and what was the price of a magazine.

Discussion Questions Day 3

DQ1: True or False? Explain your answer. The equations y = 3x - 6 and y = 2x + 4 are independent.

DQ2: Which of the following equations is not equivalent to 2x - 3y = 6?
1) 3y - 2x = 6
2)
3)
4) 2(x - 5) = 3y - 4

DQ3: Which of the following equations is inconsistent with the equation 3x + 4y = 8?
1)
2) 6x + 8y = 16
3)
4) 3x - 4y = 8

- Discussion Questions Day 5

DQ1: True or False? Explain your answer. To solve the following system of equations by addition, we can multiply the first equation by 2 and the second by 3 and then add.
3x - y = 9
2x + y = 6

DQ2: True or False? Explain your answer. The solution set to the following system of equations is infinitely many solutions.
4x - 2y = 20
-2x + y = -10

DQ3: What is an inconsistent system? What happens if you attempt to solve such a system by graphing?

DQ4: When using the addition method, how can you tell if a system of linear equations has infinitely many solutions?

DQ5: When using the substitution method, how can you tell if a system of linear equations has no solution

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